Methods of multivariate analysis
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Wiley
2002
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| Ausgabe: | 2. ed. |
| Schriftenreihe: | Wiley series in probability and statistics
A Wiley-Interscience publication |
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| 245 | 1 | 0 | |a Methods of multivariate analysis |c Alvin C. Rencher |
| 250 | |a 2. ed. | ||
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| 650 | 7 | |a Multivariate analyse |2 gtt | |
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Datensatz im Suchindex
| _version_ | 1819632379718795264 |
|---|---|
| adam_text | Titel: Methods of multivariate analysis
Autor: Rencher, Alvin C.
Jahr: 2002
Contents
1. Introduction
1.1 Why Multivariate Analysis?, I
1.2 Prerequisites, 3
1.3 Objectives, 3
1.4 Basic Types of Data and Analysis, 3
2. Matrix Algebra
2.1 Introduction, 5
2.2 Notation and Basic Definitions, 5
2.2.1 Matrices, Vectors, and Scalars, 5
2.2.2 Equality of Vectors and Matrices, 7
2.2.3 Transpose and Symmetric Matrices, 7
2.2.4 Special Matrices, 8
2.3 Operations, 9
2.3.1 Summation and Product Notation, 9
2.3.2 Addition of Matrices and Vectors, 10
2.3.3 Multiplication of Matrices and Vectors, 11
2.4 Partitioned Matrices, 20
2.5 Rank, 22
2.6 Inverse, 23
2.7 Positive Definite Matrices, 25
2.8 Determinants, 26
2.9 Trace, 30
2.10 Orthogonal Vectors and Matrices, 31
2.11 Eigenvalues and Eigenvectors, 32
2.11.1 Definition, 32
2.11.2 1 +A and I-A, 33
2.11.3 tr(A)and|A|, 34
2.11.4 Positive Definite and Semidefinite Matrices, 34
2.11.5 The Product AB, 35
2.11.6 Symmetric Matrix, 35
vi CONTENTS
2.11.7 Spectral Decomposition, 35
2.11.8 Square Root Matrix, 36
2.11.9 Square Matrices and Inverse Matrices, 36
2.11.10 Singular Value Decomposition, 36
3. Characterizing and Displaying Multivariate Data 43
3.1 Mean and Variance of a Univariate Random Variable, 43
3.2 Covariance and Correlation of Bivariate Random Variables, 45
3.2.1 Covariance, 45
3.2.2 Correlation, 49
3.3 Scatter Plots of Bivariate Samples, 50
3.4 Graphical Displays for Multivariate Samples, 52
3.5 Mean Vectors, 53
3.6 Covariance Matrices, 57
3.7 Correlation Matrices, 60
3.8 Mean Vectors and Covariance Matrices for Subsets of
Variables, 62
3.8.1 Two Subsets, 62
3.8.2 Three or More Subsets, 64
3.9 Linear Combinations of Variables, 66
3.9.1 Sample Properties, 66
3.9.2 Population Properties, 72
3.10 Measures of Overall Variability, 73
3.11 Estimation of Missing Values, 74
3.12 Distance between Vectors, 76
4. The Multivariate Normal Distribution 82
4.1 Multivariate Normal Density Function, 82
4.1.1 Univariate Normal Density, 82
4.1.2 Multivariate Normal Density, 83
4.1.3 Generalized Population Variance, 83
4.1.4 Diversity of Applications of the Multivariate Normal, 85
4.2 Properties of Multivariate Normal Random Variables, 85
4.3 Estimation in the Multivariate Normal, 90
4.3.1 Maximum Likelihood Estimation, 90
4.3.2 Distribution of y and S, 91
4.4 Assessing Multivariate Normality, 92
4.4.1 Investigating Univariate Normality, 92
4.4.2 Investigating Multivariate Normality, 96
CONTENTS VII
4.5 Outliers, 99
4.5.1 Outliers in Univariate Samples, 100
4.5.2 Outliers in Multivariate Samples, 101
5. Tests on One or Two Mean Vectors 112
5.1 Multivariate versus Univariate Tests, 112
5.2 Tests on ì with X Known, 113
5.2.1 Review of Univariate Test for Ho: ì = ìï
with ó Known, 113
5.2.2 Multivariate Test for Ho: ì = ìï with X Known, 114
5.3 Tests on ì When X Is Unknown, 117
5.3.1 Review of Univariate i-Test for Hq: ì — ìï with ó
Unknown, 117
5.3.2 Hotelling s T2-Test for HQ: ì = ìï with X Unknown, 117
5.4 Comparing Two Mean Vectors, 121
5.4.1 Review of Univariate Two-Sample/-Test, 121
5.4.2 Multivariate Two-Sample r2-Test, 122
5.4.3 Likelihood Ratio Tests, 126
5.5 Tests on Individual Variables Conditional on Rejection of Hq by
ther2-Test, 126
5.6 Computation of T2, 130
5.6.1 Obtaining 7* from a MANOVA Program, 130
5.6.2 Obtaining T2 from Multiple Regression, 130
5.7 Paired Observations Test, 132
5.7.1 Univariate Case, 132
5.7.2 Multivariate Case, 134
5.8 Test for Additional Information, 136
5.9 Profile Analysis, 139
5.9.1 One-Sample Profile Analysis, 139
5.9.2 Two-Sample Profile Analysis, 141
6. Multivariate Analysis of Variance 156
6.1 One-Way Models, 156
6.1.1 Univariate One-Way Analysis of Variance (ANOVA), 156
6.1.2 Multivariate One-Way Analysis of Variance Model
(MANOVA), 158
6.1.3 Wilks Test Statistic, 161
6.1.4 Roy s Test, 164
6.1.5 Pillai and Lawley-Hotelling Tests, 166
Viii CONTENTS
6.1.6 Unbalanced One-Way MANOVA, 168
6.1.7 Summary of the Four Tests and Relationship to T2, 168
6.1.8 Measures of Multivariate Association, 173
6.2 Comparison of the Four Manova Test Statistics, 176
6.3 Contrasts, 178
6.3.1 Univariate Contrasts, 178
6.3.2 Multivariate Contrasts, 180
6.4 Tests on Individual Variables Following Rejection of Ho by the
Overall MANOVA Test, 183
6.5 Two-Way Classification, 186
6.5.1 Review of Univariate Two-Way ANOVA, 186
6.5.2 Multivariate Two-Way MANOVA, 188
6.6 Other Models, 195
6.6.1 Higher Order Fixed Effects, 195
6.6.2 Mixed Models, 196
6.7 Checking on the Assumptions, 198
6.8 Profile Analysis, 199
6.9 Repeated Measures Designs, 204
6.9.1 Multivariate vs. Univariate Approach, 204
6.9.2 One-Sample Repeated Measures Model, 208
6.9.3 fc-Sample Repeated Measures Model, 211
6.9.4 Computation of Repeated Measures Tests, 212
6.9.5 Repeated Measures with Two Within-Subjects
Factors and One Between-Subjects Factor, 213
6.9.6 Repeated Measures with Two Within-Subjects
Factors and Two Between-Subjects Factors, 219
6.9.7 Additional Topics, 221
6.10 Growth Curves, 221
6.10.1 Growth Curve for One Sample, 221
6.10.2 Growth Curves for Several Samples, 229
6.10.3 Additional Topics, 230
6.11 Tests on a Subvector, 231
6.11.1 Test for Additional Information, 231
6.11.2 Stepwise Selection of Variables, 233
7. Tests on Covariance Matrices 248
7.1 Introduction, 248
7.2 Testing a Specified Pattern for X, 248
7.2.1 TestingH0:X = Óï, 248
CONTENTS ¡?
7.2.2 Testing Sphericity, 250
7.2.3 Testing Ho : S = s2[(1 - p)I + pi], 252
7.3 Tests Comparing Covariance Matrices, 254
7.3.1 Univariate Tests of Equality of Variances, 254
7.3.2 Multivariate Tests of Equality of Covariance Matrices, 255
7.4 Tests of Independence, 259
7.4.1 Independence of Two Subvectors, 259
7.4.2 Independence of Several Subvectors, 261
7.4.3 Test for Independence of All Variables, 265
8. Discriminant Analysis: Description of Group Separation 270
8.1 Introduction, 270
8.2 The Discriminant Function for Two Groups, 271
8.3 Relationship between Two-Group Discriminant Analysis and
Multiple Regression, 275
8.4 Discriminant Analysis for Several Groups, 277
8.4.1 Discriminant Functions, 277
8.4.2 A Measure of Association for Discriminant Functions, 282
8.5 Standardized Discriminant Functions, 282
8.6 Tests of Significance, 284
8.6.1 Tests for the Two-Group Case, 284
8.6.2 Tests for the Several-Group Case, 285
8.7 Interpretation of Discriminant Functions, 288
8.7.1 Standardized Coefficients, 289
8.7.2 Partial F-Values, 290
8.7.3 Correlations between Variables and Discriminant
Functions, 291
8.7.4 Rotation, 291
8.8 Scatter Plots, 291
8.9 Stepwise Selection of Variables, 293
9. Classification Analysis: Allocation of Observations to Groups 299
9.1 Introduction, 299
9.2 Classification into Two Groups, 300
9.3 Classification into Several Groups, 304
9.3.1 Equal Population Covariance Matrices: Linear
Classification Functions, 304
9.3.2 Unequal Population Covariance Matrices: Quadratic
Classification Functions, 306
X CONTENTS
9.4 Estimating Misclassification Rates, 307
9.5 Improved Estimates of Error Rates, 309
9.5.1 Partitioning the Sample, 310
9.5.2 Holdout Method, 310
9.6 Subset Selection, 311
9.7 Nonparametric Procedures, 314
9.7.1 Multinomial Data, 314
9.7.2 Classification Based on Density Estimators, 315
9.7.3 Nearest Neighbor Classification Rule, 318
10. Multivariate Regression 322
10.1 Introduction, 322
10.2 Multiple Regression: Fixed x s, 323
10.2.1 Model for Fixed ÷ s, 323
10.2.2 Least Squares Estimation in the Fixed-* Model, 324
10.2.3 An Estimator for ó2, 326
10.2.4 The Model Corrected for Means, 327
10.2.5 Hypothesis Tests, 329
10.2.6 R2 in Fixed-* Regression, 332
10.2.7 Subset Selection, 333
10.3 Multiple Regression: Random ÷ s, 337
10.4 Multivariate Multiple Regression: Estimation, 337
10.4.1 The Multivariate Linear Model, 337
10.4.2 Least Squares Estimation in the Multivariate Model, 339
10.4.3 Properties of Least Squares Estimators B, 341
10.4.4 An Estimator for 2, 342
10.4.5 Model Corrected for Means, 342
10.5 Multivariate Multiple Regression: Hypothesis Tests, 343
10.5.1 Test of Overall Regression, 343
10.5.2 Test on a Subset of the * s, 347
10.6 Measures of Association between the y sand them s, 349
10.7 Subset Selection, 351
10.7.1 Stepwise Procedures, 351
10.7.2 All Possible Subsets, 355
10.8 Multivariate Regression: Random jc s, 358
11. Canonical Correlation 361
11.1 Introduction, 361
11.2 Canonical Correlations and Canonical Variates, 361
CONTENTS xi
11.3 Properties of Canonical Correlations, 366
11.4 Tests of Significance, 367
11.4.1 Tests of No Relationship between the y s and the ?: s, 367
11.4.2 Test of Significance of Succeeding Canonical
Correlations after the First, 369
11.5 Interpretation, 371
11.5.1 Standardized Coefficients, 371
11.5.2 Correlations between Variables and Canonical Variâtes, 373
11.5.3 Rotation, 373
11.5.4 Redundancy Analysis, 373
11.6 Relationships of Canonical Correlation Analysis to Other
Multivariate Techniques, 374
11.6.1 Regression, 374
11.6.2 MANOVA and Discriminant Analysis, 376
12. Principal Component Analysis 380
12.1 Introduction, 380
12.2 Geometric and Algebraic Bases of Principal Components, 381
12.2.1 Geometric Approach, 381
12.2.2 Algebraic Approach, 385
12.3 Principal Components and Perpendicular Regression, 387
12.4 Plotting of Principal Components, 389
12.5 Principal Components from the Correlation Matrix, 393
12.6 Deciding How Many Components to Retain, 397
12.7 Information in the Last Few Principal Components, 401
12.8 Interpretation of Principal Components, 401
12.8.1 Special Patterns in S or R, 402
12.8.2 Rotation, 403
12.8.3 Correlations between Variables and Principal
Components, 403
12.9 Selection of Variables, 404
13. Factor Analysis 408
13.1 Introduction, 408
13.2 Orthogonal Factor Model, 409
13.2.1 Model Definition and Assumptions, 409
13.2.2 Nonuniqueness of Factor Loadings, 414
13.3 Estimation of Loadings and Communalities, 415
13.3.1 Principal Component Method, 415
13.3.2 Principal Factor Method, 421
XU CONTENTS
13.3.3 Iterated Principal Factor Method, 424
13.3.4 Maximum Likelihood Method, 425
13.4 Choosing the Number of Factors, m, 426
13.5 Rotation, 430
13.5.1 Introduction, 430
13.5.2 Orthogonal Rotation, 431
13.5.3 Oblique Rotation, 435
13.5.4 Interpretation, 438
13.6 Factor Scores, 438
13.7 Validity of the Factor Analysis Model, 443
13.8 The Relationship of Factor Analysis to Principal Component
Analysis, 447
14. Cluster Analysis 451
14.1 Introduction, 451
14.2 Measures of Similarity or Dissimilarity, 452
14.3 Hierarchical Clustering, 455
14.3.1 Introduction, 455
14.3.2 Single Linkage (Nearest Neighbor), 456
14.3.3 Complete Linkage (Farthest Neighbor), 459
14.3.4 Average Linkage, 463
14.3.5 Centroid, 463
14.3.6 Median, 466
14.3.7 Ward s Method, 466
14.3.8 Flexible Beta Method, 468
14.3.9 Properties of Hierarchical Methods, 471
14.3.10 Divisive Methods, 479
14.4 Nonhierarchical Methods, 481
14.4.1 Partitioning, 481
14.4.2 Other Methods, 490
14.5 Choosing the Number of Clusters, 494
14.6 Cluster Validity, 496
14.7 Clustering Variables, 497
15. Graphical Procedures 504
15.1 Multidimensional Scaling, 504
15.1.1 Introduction, 504
15.1.2 Metric Multidimensional Scaling, 505
15.1.3 Nonmetric Multidimensional Scaling, 508
CONTENTS Xiit
15.2 Correspondence Analysis, 514
15.2.1 Introduction, 514
15.2.2 Row and Column Profiles, 515
15.2.3 Testing Independence, 519
15.2.4 Coordinates for Plotting Row and Column Profiles, 521
15.2.5 Multiple Correspondence Analysis, 526
15.3 Biplots, 531
15.3.1 Introduction, 531
15.3.2 Principal Component Plots, 531
15.3.3 Singular Value Decomposition Plots, 532
15.3.4 Coordinates, 533
15.3.5 Other Methods, 535
A. Tables 549
B. Answers and Hints to Problems 591
C. Data Sets and SAS Files 679
References 681
Index 695
|
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV023528251 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T21:16:38Z |
| institution | BVB |
| isbn | 0471418897 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016848454 |
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| physical | XXII, 708 S. graph. Darst. |
| publishDate | 2002 |
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| publisher | Wiley |
| record_format | marc |
| series2 | Wiley series in probability and statistics A Wiley-Interscience publication |
| spellingShingle | Rencher, Alvin C. 1934- Methods of multivariate analysis Analyse multivariée Analyse multivariée ram Análise multivariada larpcal Multivariate analyse gtt Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| subject_GND | (DE-588)4040708-1 (DE-588)4123623-3 |
| title | Methods of multivariate analysis |
| title_auth | Methods of multivariate analysis |
| title_exact_search | Methods of multivariate analysis |
| title_full | Methods of multivariate analysis Alvin C. Rencher |
| title_fullStr | Methods of multivariate analysis Alvin C. Rencher |
| title_full_unstemmed | Methods of multivariate analysis Alvin C. Rencher |
| title_short | Methods of multivariate analysis |
| title_sort | methods of multivariate analysis |
| topic | Analyse multivariée Analyse multivariée ram Análise multivariada larpcal Multivariate analyse gtt Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| topic_facet | Analyse multivariée Análise multivariada Multivariate analyse Multivariate analysis Multivariate Analyse Lehrbuch |
| url | http://www.loc.gov/catdir/description/wiley0310/2001046735.html http://www.loc.gov/catdir/toc/wiley021/2001046735.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016848454&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rencheralvinc methodsofmultivariateanalysis |